LMFDB - Newform orbit 1440.1.bh.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,1,Mod(577,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.577"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$1440 = 2^{5} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1440.bh (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.718653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.18000.1

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q + q^{5} + (i + 1) q^{13} + (i - 1) q^{17} + q^{25} - 2 i q^{29} + ( - i + 1) q^{37} - 2 q^{41} + i q^{49} + (i + 1) q^{53} + (i + 1) q^{65} + ( - i - 1) q^{73} + (i - 1) q^{85} - 2 i q^{89} + (i - 1) q^{97} +O(q^{100})$ q + q^5 + (z + 1) * q^13 + (z - 1) * q^17 + q^25 - 2z q^29 + (-z + 1) * q^37 - 2 * q^41 + z * q^49 + (z + 1) * q^53 + (z + 1) * q^65 + (-z - 1) * q^73 + (z - 1) * q^85 - 2z q^89 + (z - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 2 q^{13} - 2 q^{17} + 2 q^{25} + 2 q^{37} - 4 q^{41} + 2 q^{53} + 2 q^{65} - 2 q^{73} - 2 q^{85} - 2 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 + 2 * q^13 - 2 * q^17 + 2 * q^25 + 2 * q^37 - 4 * q^41 + 2 * q^53 + 2 * q^65 - 2 * q^73 - 2 * q^85 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$ .

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-i$	$1$	$1$	$1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

577.1

	−	1.00000i
		1.00000i

1.00000

1153.1

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1440.1.bh.c	yes	2
3.b	odd	2	1		1440.1.bh.a	✓	2
4.b	odd	2	1	CM	1440.1.bh.c	yes	2
5.c	odd	4	1	inner	1440.1.bh.c	yes	2
8.b	even	2	1		2880.1.bh.a		2
8.d	odd	2	1		2880.1.bh.a		2
12.b	even	2	1		1440.1.bh.a	✓	2
15.e	even	4	1		1440.1.bh.a	✓	2
20.e	even	4	1	inner	1440.1.bh.c	yes	2
24.f	even	2	1		2880.1.bh.c		2
24.h	odd	2	1		2880.1.bh.c		2
40.i	odd	4	1		2880.1.bh.a		2
40.k	even	4	1		2880.1.bh.a		2
60.l	odd	4	1		1440.1.bh.a	✓	2
120.q	odd	4	1		2880.1.bh.c		2
120.w	even	4	1		2880.1.bh.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1440.1.bh.a	✓	2	3.b	odd	2	1
1440.1.bh.a	✓	2	12.b	even	2	1
1440.1.bh.a	✓	2	15.e	even	4	1
1440.1.bh.a	✓	2	60.l	odd	4	1
1440.1.bh.c	yes	2	1.a	even	1	1	trivial
1440.1.bh.c	yes	2	4.b	odd	2	1	CM
1440.1.bh.c	yes	2	5.c	odd	4	1	inner
1440.1.bh.c	yes	2	20.e	even	4	1	inner
2880.1.bh.a		2	8.b	even	2	1
2880.1.bh.a		2	8.d	odd	2	1
2880.1.bh.a		2	40.i	odd	4	1
2880.1.bh.a		2	40.k	even	4	1
2880.1.bh.c		2	24.f	even	2	1
2880.1.bh.c		2	24.h	odd	2	1
2880.1.bh.c		2	120.q	odd	4	1
2880.1.bh.c		2	120.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1440, [\chi])$ :

T_{13}^{2} - 2T_{13} + 2

T13^2 - 2*T13 + 2

T_{17}^{2} + 2T_{17} + 2

T17^2 + 2*T17 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$T^{2}$ T^2
$11$	$T^{2}$ T^2
$13$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$17$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2} + 4$ T^2 + 4
$31$	$T^{2}$ T^2
$37$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$41$	$(T + 2)^{2}$ (T + 2)^2
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2} - 2T + 2$ T^2 - 2*T + 2
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
$79$	$T^{2}$ T^2
$83$	$T^{2}$ T^2
$89$	$T^{2} + 4$ T^2 + 4
$97$	$T^{2} + 2T + 2$ T^2 + 2*T + 2
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